Optimal. Leaf size=116 \[ -\frac {2^{-\frac {n}{2}+p+1} \left (1-\frac {1}{a^2 x^2}\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \left (\frac {1}{a x}+1\right )^{\frac {n}{2}+p+1} F_1\left (\frac {n}{2}+p+1;\frac {1}{2} (n-2 p),2;\frac {n}{2}+p+2;\frac {a+\frac {1}{x}}{2 a},1+\frac {1}{a x}\right )}{a (n+2 p+2)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.11, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6197, 6194, 136} \[ -\frac {2^{-\frac {n}{2}+p+1} \left (1-\frac {1}{a^2 x^2}\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \left (\frac {1}{a x}+1\right )^{\frac {n}{2}+p+1} F_1\left (\frac {n}{2}+p+1;\frac {1}{2} (n-2 p),2;\frac {n}{2}+p+2;\frac {a+\frac {1}{x}}{2 a},1+\frac {1}{a x}\right )}{a (n+2 p+2)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 136
Rule 6194
Rule 6197
Rubi steps
\begin {align*} \int e^{n \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx &=\left (\left (1-\frac {1}{a^2 x^2}\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p\right ) \int e^{n \coth ^{-1}(a x)} \left (1-\frac {1}{a^2 x^2}\right )^p \, dx\\ &=-\left (\left (\left (1-\frac {1}{a^2 x^2}\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p\right ) \operatorname {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{-\frac {n}{2}+p} \left (1+\frac {x}{a}\right )^{\frac {n}{2}+p}}{x^2} \, dx,x,\frac {1}{x}\right )\right )\\ &=-\frac {2^{1-\frac {n}{2}+p} \left (1-\frac {1}{a^2 x^2}\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \left (1+\frac {1}{a x}\right )^{1+\frac {n}{2}+p} F_1\left (1+\frac {n}{2}+p;\frac {1}{2} (n-2 p),2;2+\frac {n}{2}+p;\frac {a+\frac {1}{x}}{2 a},1+\frac {1}{a x}\right )}{a (2+n+2 p)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [F] time = 0.59, size = 0, normalized size = 0.00 \[ \int e^{n \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n} \left (\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}\right )^{p}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{p} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.13, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{n \,\mathrm {arccoth}\left (a x \right )} \left (c -\frac {c}{a^{2} x^{2}}\right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{p} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\mathrm {e}}^{n\,\mathrm {acoth}\left (a\,x\right )}\,{\left (c-\frac {c}{a^2\,x^2}\right )}^p \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (- c \left (-1 + \frac {1}{a x}\right ) \left (1 + \frac {1}{a x}\right )\right )^{p} e^{n \operatorname {acoth}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________